Wellposedness of Cauchy problem for the Fourth Order Nonlinear Schr\"odinger Equations in Multi-dimensional Spaces

TL;DR

This paper investigates the mathematical conditions under which the initial value problem for a fourth order nonlinear Schrödinger equation is well-posed in multi-dimensional spaces, ensuring solutions exist, are unique, and depend continuously on initial data.

Contribution

The study establishes well-posedness results for the fourth order nonlinear Schrödinger equations in higher dimensions, extending previous understanding to more complex multi-dimensional cases.

Findings

Proved well-posedness in multi-dimensional spaces for certain initial data.

Identified conditions on the polynomial nonlinearity for solution existence.

Extended the theory to include equations with various sign parameters ps.

Abstract

We study the wellposedness of Cauchy problem for the fourth order nonlinear Schr\"odinger equations i\partial_t u=-\eps\Delta u+\Delta^2 u+P((\partial_x^\alpha u)_{\abs{\alpha}\ls 2}, (\partial_x^\alpha \bar{u})_{\abs{\alpha}\ls 2}),\quad t\in \Real, x\in\Real^n, where $\eps\in\{-1,0,1\}$, $n\gs 2$ denotes the spatial dimension and $P(\cdot)$ is a polynomial excluding constant and linear terms.